Image analysis with logarithmic Zernike moments

As one of the most widely used orthogonal moments, Zernike moments (ZMs) have been applied in various fields. But low-order ZMs encounter problem for describing small size images. The cause of this problem lies in that zeros of ZMs' radial basis function (RBF) bias toward large radial distance from the origin. Fractional Zernike moments (FrZMs) can be utilized to deal with small size images by adjusting RBF's zeros distribution with fractional-order parameter. But high-order FrZMs will result in numerical instability. To address these problems, ZMs are generalized to transformed Zernike moments (TZMs) and logarithmic Zernike moments (LoZMs) are proposed. ZMs and FrZMs are only special cases of TZMs. LoZMs are constructed by a particularly designed transformed function such that zeros of LoZMs'RBF are near evenly distributed in interval [0,1]. As a result, small size images can be well represented with low-order LoZMs. Furthermore, amplitudes of LoZMs'RBF keep relatively stable and numerical instability of high-order moments is suppressed. Extensive experiments are conducted to demonstrate the superior performance of LoZMs in image reconstruction and pattern recognition tasks. In addition, a quaternion-based zero-watermarking algorithm also shows that LoZMs perform better than ZMs.